The bounding number,  is one of the cardinal characteristics of the continuum. Here, we begin to study a generalized version of the bounding number.

For a set A, let  denote the set of functions of A into A. Let  be a cardinal. For  we write  when the cardinality of the set  is less than  The bounding number  for  is defined as the minimal cardinality of a -unbounded subset of  This definition generalizes the definition of the bounding number  which equals in our notation to

In this paper, we prove the inequality  for each cardinal

bounding number, generalized reals, cardinal characteristics of the continuum, infinite combinatorics.